Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general technique to lift results from the smooth case to the essentially smooth case?

For instance, if $R$ is reduced, and $R\to S$ is smooth, then $S$ is reduced. It turns out that the same fact is true for essentially smooth maps. Is there a good way to "lift" this result to the more general case?

Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general technique to lift results from the smooth case to the essentially smooth case?

Edit: According to Mel, every essentially smooth morphism is a localization of a smooth morphism. However, this direction is much more involved than the other direction, which is immediate from the definitions. Anyway, this would be the answer to the question.

Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general way to lift results from the smooth case to the essentially smooth case?

For instance, if $R$ is reduced, and $R\to S$ is smooth, then $S$ is reduced. It turns out that the same fact is true for essentially smooth maps. Is there a good way to "lift" this result to the more general case?

Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general technique to lift results from the smooth case to the essentially smooth case?

For instance, if $R$ is reduced, and $R\to S$ is smooth, then $S$ is reduced. It turns out that the same fact is true for essentially smooth maps. Is there a good way to "lift" this result to the more general case?

A morphism of rings $R\to S$ is called essentially smooth if it is formally smooth and essentially finitely presented, where essentially finitely presented means that $S$ is the localization of some finitely presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.

Can we factor an essentially smooth map into a smooth map and a localization?
Certainly the converse is true, i.e. the localization of any smooth $R$-algebra is essentially smooth, since localizations are formally étale (except of course when the multiplicative system contains zero.)

If not, are there any conditions for when we can?

Recall that a morphism of rings $R\to S$ is called (essentially) smooth if it is formally smooth and (essentially) finitely presented.

(Note: $R\to S$ is essentially finitely presented provided that $S$ is the localization of some finitely
presented $R$-algebra $T$ at some multiplicative system $A \subset T$, that is, $S=A^{-1}T$.)

In class, our professor said that working with smooth or essentially smooth morphisms yields an effectively equivalent theory. This motivates my question: Is there a general way to lift results from the smooth case to the essentially smooth case?

For instance, if $R$ is reduced, and $R\to S$ is smooth, then $S$ is reduced. It turns out that the same fact is true for essentially smooth maps. Is there a good way to "lift" this result to the more general case?